WAEC 2018 · Paper 2 · Q1

(a)
If $x = 2 + \sqrt5$ , find the value of $x - \dfrac1x$ and leave the answer in the form $m + n\sqrt5$ .

Worked solution (try it first)

Rationalise $\frac1x$ : multiply top and bottom by $\sqrt5 - 2$ .
The bottom is $(2 + \sqrt5)(\sqrt5 - 2) = 5 - 4 = 1$ , so $\dfrac1x = \sqrt5 - 2$ .
Subtract: $x - \dfrac1x = (2 + \sqrt5) - (\sqrt5 - 2)$
$= 4$ .
In the form $m + n\sqrt5$ , this is $4 + 0\sqrt5$ .